In immersed finite element methods the geometry representation is independent of the mesh. In general, there are two ways to represent the geometry, viz. implicit representations (e.g., voxel data [1]}, [2]} or a level set function [3]}) and explicit boundary representations (e.g., spline surfaces [4]}, B-rep objects [5]}, or isogeometric analysis on trimmed CAD objects [6]}, [7]}). For all geometry representations, dedicated techniques are required to evaluate volume integrals over the intersection of active elements with the domain. A myriad of such integration procedures has been developed over the years in the context of immersed FEM (see [8]}, [9]} for reviews/comparisons) and enriched FEM (see [10]}), an overview of which is presented below. Techniques to integrate over trimmed boundaries are strongly dependent on the geometry description. In explicit representations, the geometry description itself can sometimes be leveraged, whereas generally a boundary-reconstruction procedure is required in the case of implicit boundary representations. The reader is referred to, e.g., Refs. [9]} for a discussion regarding the various techniques to handle different geometry representations, and the related problem of integrating over unfitted boundaries.
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